Mathematics

When I multiply 2 by 2, is it by a form of reasoning that I produce the result, or rather mere memorization? Does the same hold for multiplications of larger numbers, or arithmetic operations generally?

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When an elementary-school student is learning how to multiply, the result of multiplying 2 by 2 is probably produced by a form of reasoning (perhaps repeated addition). When you or I do it, it's probably done by rote memory. But when any of us multiply two 6-digit numbers, it's almost certainly by "reasoning". (Maybe those with "savant syndrome" do it by some kind of memory-like process, or maybe it's just by very fast, unconscious reasoning.)

But the "reasoning" we use for multiplying those larger numbers consists of applying the multiplication algorithm, among whose steps are instructions to multiply single-digit numbers together (like 2x2, or 9x8). And those multiplications are probably done by "memory" (what computer scientists call "table look-up").

That's because multiplication is a recursive procedure: We multiply large numbers by applying the multiplication algorithm, which requires us to multiply smaller numbers, eventually "bottoming out" in the base case of table-look up of the product of two single-digit numbers.

But let's return to what the elementary-school student is doing who is learning to multiply by the "reasoning" process of repeated addition. The addition algorithm is similarly recursive: We add large numbers by applying the addition algorithm to smaller numbers, bottoming out with the addition of two single-digit numbers. The result of adding two single-digit numbers is probably memorized, but could be "reasoned" out by a process like "adding on": To add 5 and 2, start with 5 and count up for two numbers: 5, 6, 7. And this process itself is recursive, bottoming out with "adding 1", i.e., finding the successor of a number.

So, to answer your question using your terminology, I'd say that we multiply large numbers by what you're calling a reasoning process, but the "basic level" of that reasoning process is probably what you're calling a memorization process.

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